We give upper and lower bounds for the Kazhdan-Lusztig polynomials of any Coxeter group W. If W is nite we prove that, for any k 0, the k-th coeecient of the Kazhdan-Lusztig polynomial of two elements u, v of W is bounded from above and below by a polynomial (which depends only on k) in l(v)?l(u). In particular, this implies the validity of Lascoux-Schutzenberger's conjecture for all suuciently...

Let U be the quantum group associated to a Lie algebra g of type An. The negative part U of U has a canonical basis B defined by Lusztig and Kashiwara, with favourable properties. We show how the spanning vectors of the cones defined by Lusztig [12], when regarded as monomials in Kashiwara’s root operators, can be described using a remarkable rectangle combinatorics. We use this to calculate th...

We study 2-representations of finitary 2-categories with involution and adjunctions by functors on module categories over finite dimensional algebras. In particular, we define, construct and describe in detail (right) cell 2-representations inspired by KazhdanLusztig cell modules for Hecke algebras. Under some natural assumptions we show that cell 2-representations are strongly simple and do no...

Lusztig has given a construction of certain representations of reductive groups over finite local principal ideal rings of characteristic p, extending the construction of Deligne and Lusztig of representations of reductive groups over finite fields. We generalize Lusztig’s results to reductive groups over arbitrary finite local rings. This generalization uses the Greenberg functor and the theor...

Lusztig proved that the Kazhdan-Lusztig basis of a spherical algebra can be essentially identified with the Weyl characters of the Langlands dual group. We generalize this result to the unequal parameter case. Our new proof is simple and quite different from Lusztig’s.

We prove a nonsymmetric analogue of a formula of Kato and Lusztig which describes the coefficients of the expansion of irreducible Weyl characters in terms of (degenerate) symmetric Macdonald polynomials as certain Kazhdan–Lusztig polynomials. We also establish precise polynomiality results for coefficients of symmetric and nonsymmetric Macdonald polynomials and a version of Demazure’s characte...

We find an explicit formula for the Kazhdan-Lusztig polynomials Pui,a ,vi of the symmetric group S(n) where, for a, i, n ∈ N such that 1 ≤ a ≤ i ≤ n, we denote by ui,a = sasa+1 · · · si−1 and by vi the element ofS(n) obtained by inserting n in position i in any permutation ofS(n −1) allowed to rise only in the first and in the last place. Our result implies, in particular, the validity of two c...

We use a quantum analog of the polynomial ringZ[x1,1, . . . , xn,n] to modify the Kazhdan-Lusztig construction of irreducible Hn(q)-modules. This modified construction produces exactly the same matrices as the original construction in [Invent. Math. 53 (1979)], but does not employ the Kazhdan-Lusztig preorders. Our main result is dependent on new vanishing results for immanants in the quantum p...

A survey is given on the achievements of KL-cell theory of certain Coxeter groups. Some techniques applied in that theory are introduced. Also, we propose several open problems for further study. In order to construct the representations of a Coxeter group W and the associative Hecke algebra, D. Kazhdan and G. Lusztig defined certain equivalence classes of W called left, right and two-sided cel...